CSE332: Data Abstrac0ons Sec%on 2. HyeIn Kim Spring 2014

Transcription

1 CSE332: Data Abstrac0ons Sec%on 2 HyeIn Kim Spring 2014

2 Sec0on Agenda Recurrence Relations Asymptotic Analysis HW1, Project 1 Q&A Project 2 - Introduction - Working in a team - Testing strategies

3 Recurrence Rela0ons

4 Recurrence Rela0ons Recursively defines a Sequence - Example: T(n) = T(n-1) + 3, T(1) = 5 ^ Has T(x) in definition Solving Recurrence Relation - Eliminate recursive part in definition = Find Closed Form - Example: T(n) = 3n + 2

5 Recurrence Rela0ons Expansion Method example - Solve T(n) = T(n-1) + 2n 1, T(1) = 1 T(n) = T(n-1) + 2n 1 T(n-1) = T([n-1]-1) + 2[n-1] 1 = T(n-2) + 2(n-1) 1 T(n-2) = T([n-2]-1) + 2[n-2] 1 = T(n-3) + 2(n-2) 1

6 Recurrence Rela0ons Expansion Method example T(n) = T(n-1) + 2n 1 T(n-1) = T(n-2) + 2(n-1) 1 T(n-2) = T(n-3) + 2(n-2) 1 T(n) = [T(n-2) + 2(n-1) 1] + 2n 1 = T(n-2) + 2(n-1) + 2n 2 T(n) = [T(n-3) + 2(n-2) 1] + 2(n-1) + 2n 2 = T(n-3) + 2(n-2) + 2(n-1) + 2n 3

7 Recurrence Rela0ons Expansion Method example T(n) = T(n-1) + 2n 1 T(n) = T(n-2) + 2(n-1) + 2n 2 T(n) = T(n-3) + 2(n-2) + 2(n-1) + 2n 3 T(n) = T(n-k) + [2(n-(k-1)) + + 2(n-1) + 2n] k = T(n-k) + [2(n-k+1) + + 2(n-1) + 2n] k

8 Recurrence Rela0ons Expansion Method example T(n) = T(n-k) + [2(n-k+1) + + 2(n-1) + 2n] k When expanded all the way down, T(n-k) = T(1) n-k = 1, k = n-1 T(n) = T(n-[n-1]) + [2(n-[n-1]+1) + + 2(n-1) + 2n] [n-1] = T(1) + [2(2) + + 2(n-1) + 2n] n + 1

9 Recurrence Rela0ons Expansion Method example T(n) = T(1) + [2(2) + + 2(n-1) + 2n] n + 1 = T(1) + 2[2 + + (n-1) + n] n + 1 = T(1) + 2[(n+1)(n/2) -1] n + 1 = T(1) + (n+1)(n) - 2 n + 1 = T(1) + (n 2 +n) n - 1 = T(1) + n 2 1 = 1 + n 2 1 = n 2

10 Recurrence Rela0ons Expansion Method example Check it! T(n) = T(n-1) + 2n 1, T(1) = 1 T(n) = n 2 T(1) = 1 same as 1 2 T(2) = T(1) + 2(2) 1 = 4 same as 2 2 T(3) = T(2) + 2(3) 1 = 9 same as 3 2 T(4) = T(3) + 2(4) 1 = 16 same as 4 2

11 Recurrence Rela0ons For Homework Remember to show steps!! - Correct answer with no steps gets no credit a) Show at least 2 expansions of T(n) b) At least 2 representations of T(n), using a) c) Writing T(n) in terms of k, using b) d) Solve for k (show steps!!) e) Plug in k and get the closed form

12 Asympto0c Analysis

13 Asympto0c Analysis Describe Limiting behavior of F(n) - Characterize growth rate of F(n) - Use O(g(n)), Ω(g(n)), Θ(g(n)) for set of functions with asymptotic behavior,, & to g(n) Upper Bound: O(n) f(n) O(g(n)) if and only if there exist positive constants c and n 0 such that f(n) c*g(n) for all n 0 n log n O(n)

14 Asympto0c Analysis Lower Bound: Ω(n) f(n) Ω(g(n)) if and only if there exist positive constants c and n 0 such that c*g(n) f(n) for all n 0 n Tight Bound: Θ(n) f(n) Θ(g(n)) if and only if f(n) Ω(g(n)) and f(n) O(g(n)) n Ω(log n) 5*log 10 n Θ(log n)

15 Asympto0c Analysis Ordering Growth rates (k = constant) - Ignore Low-Order terms & Coefficients O(k) constant O(log n) logarithmic Increasing O(n) linear Growth rate O(n k ) polynomial O(k n ) exponential (k > 1)

16 Asympto0c Analysis Ordering Growth rates

17 Asympto0c Analysis Ordering Growth rates (k, b = constant) - log k n O(n b ) if 1 < k & 0 < b - n k O(b n ) if 0 < k & 1 < b Ordering Example 2n n 2 n/ n/ n + log 8 n 23785n 1/ log 10 n + 1 n/300 n n/100 n n 1/ log 10 n 1

18 Asympto0c Analysis Proof Example 1: f(n) O(g(n)) - Prove or disprove nlog n O(3n) nlog n O(3n), then by definition of Big-O nlog n c*(3n), for 0 < c && 0 < n 0 n (1/3)log n c but as n, log n Finite constant c always greater than log n cannot exist, no matter what n 0 we choose nlog n O(3n)

19 Asympto0c Analysis Proof Example 2 - Prove or disprove If f(n) O(g(n)) and h(n) O(k(n)), then f(n) + h(n) O(g(n) + k(n))

20 Homework 1 Q&A Solutions are NOT posted - Any questions?

21 Project 2 shake- n- bacon

22 Project 2 Word Frequency Analysis Phase A: Implement 3 ADTs - Due about 2 weeks from now (Thursday April 24 th ) - Word frequency analysis using different DataCounters AVLTree MoveToFrontList FourHeap Heap Sort DataCount data: Apple count: 21 DataCount data: Hamlet count: 67 DataCount data: water count: 33

23 Project 2 Find Partner Form a 2 person team Not required, but greatly encouraged - Learn from each other - Check each other s style (large fraction of grade!) - Collaborate working experience (You can put it on your resume!) Many had hard time meeting deadline - Don t be spoiled by project 1!

24 Project 2 Find Partner Form a 2 person team - Use discussion board to find partner - Complete catalyst survey to form team by next Friday April 18 th (Only one survey for a team!) - Anyone wants a partner but don t have one yet?

25 Project 2 Sharing Code Version Control System - Git, Mercurial, SVN, CVS, Perforce... - Lots of choices (Use whatever you want): Some posts to help choosing one - Choosing a Version Control: Beginners Tour - Review of 7 Version Control Systems - What is your favorite version control system?

26 Project 2 Sharing Code Repository: Where you store your code - Your machine - Shared directory in attu (you need to request one) - Web based repository (should be private!) Bitbucket: GitHub:

27 Project 2 Sharing Code Useful tutorials - Bitbucket Git - EGit (Git with Eclipse) Easy & Quick Code Sharing - If you don t want to bother having version control